Subgroup ($H$) information
| Description: | $C_2^4:S_3^2$ |
| Order: | \(576\)\(\medspace = 2^{6} \cdot 3^{2} \) |
| Index: | \(27\)\(\medspace = 3^{3} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(1,6)(2,8)(3,5)(4,7), (1,7)(2,5)(3,8)(4,6)(12,17,19)(15,18,16), (13,20,14) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Ambient group ($G$) information
| Description: | $C_6^4:D_6$ |
| Order: | \(15552\)\(\medspace = 2^{6} \cdot 3^{5} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_2\times C_6^3).C_3^4.C_2^4$ |
| $\operatorname{Aut}(H)$ | $(C_2^3\times S_3^2).C_2^4$ |
| $W$ | $D_6^2:C_2$, of order \(288\)\(\medspace = 2^{5} \cdot 3^{2} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $81$ |
| Number of conjugacy classes in this autjugacy class | $3$ |
| Möbius function | $0$ |
| Projective image | $C_6^4:D_6$ |